1. Field of the Invention
The present invention relates to techniques for performing arithmetic operations involving interval operands within a computer system. More specifically, the present invention relates to a method and an apparatus for solving overdetermined systems of interval linear equations within a computer system.
2. Related Art
Rapid advances in computing technology make it possible to perform trillions of computational operations each second. This tremendous computational speed makes it practical to perform computationally intensive tasks as diverse as predicting the weather and optimizing the design of an aircraft engine. Such computational tasks are typically performed using machine-representable floating-point numbers to approximate values of real numbers. (For example, see the Institute of Electrical and Electronics Engineers (IEEE) standard 754 for binary floating-point numbers.)
In spite of their limitations, floating-point numbers are generally used to perform most computational tasks.
One limitation is that machine-representable floating-point numbers have a fixed-size word length, which limits their accuracy. Note that a floating-point number is typically encoded using a 32, 64 or 128-bit binary number, which means that there are only 232, 264 or 2128 possible symbols that can be used to specify a floating-point number. Hence, most real number values can only be approximated with a corresponding floating-point number. This creates estimation errors that can be magnified through even a few computations, thereby adversely affecting the accuracy of a computation.
A related limitation is that floating-point numbers contain no information about their accuracy. Most measured data values include some amount of error that arises from the measurement process itself. This error can often be quantified as an accuracy parameter, which can subsequently be used to determine the accuracy of a computation. However, floating-point numbers are not designed to keep track of accuracy information, whether from input data measurement errors or machine rounding errors. Hence, it is not possible to determine the accuracy of a computation by merely examining the floating-point number that results from the computation.
Interval arithmetic has been developed to solve the above-described problems. Interval arithmetic represents numbers as intervals specified by a first (left) endpoint and a second (right) endpoint. For example, the interval [a, b], where a<b, is a closed, bounded subset of the real numbers, R, which includes a and b as well as all real numbers between a and b. Arithmetic operations on interval operands (interval arithmetic) are defined so that interval results always contain the entire set of possible values. The result is a mathematical system for rigorously bounding numerical errors from all sources, including measurement data errors, machine rounding errors and their interactions. (Note that the first endpoint normally contains the “infimum”, which is the largest number that is less than or equal to each of a given set of real numbers. Similarly, the second endpoint normally contains the “supremum”, which is the smallest number that is greater than or equal to each of the given set of real numbers. Also note that the infimum and the supremum can be represented by floating point numbers.)
One commonly performed computational operation is to solve a system of linear equations Ax=b, wherein A is an (n×n) matrix and b is a (n×1) column vector. Such a system is said to be “consistent” if there is a unique (n×1) vector x for which the system Ax=b is satisfied. In many cases, a system of linear equations is “overdetermined,” which means that there are more equations than unknowns. In an overdetermined system Ax=b, the number of rows in A and elements in b is m, which is greater than n (the number of columns in A and elements in x).
In the point (non-interval) case, there is no generally reliable way to decide if an overdetermined system is consistent or not. Instead a least squares solution is generally sought. However, in an overdetermined system of linear equations with interval coefficients, the additional equations can potentially help in bounding the set of solutions.
Hence, what is needed is a method and an apparatus for solving an overdetermined system of interval linear equations.